DCA6103 foundation of Mathematics

Description

SESSION	February/MARCH 2024
PROGRAM	Master of CoMPUTER APPLICATIONS (MCA)
SEMESTER	I
course CODE & NAME	DCA6103 – Foundation of Mathematics
CREDITS	4
nUMBER OF ASSIGNMENTS & Marks	02 Sets & 30 Marks

 

Set-Ist

 

1. Check whether the following is Tautology or Contradiction:
2. (p∨q)∨(∼p)

Ans: To determine whether the given logical expressions are tautologies or contradictions, we need to evaluate their truth values for all possible combinations of truth values of their constituent propositions (p and q).

 

Let’s evaluate each Its Half solved only

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2. Find the derivative of f(x)=x^2+2x using the first principle. State inclusion-exclusion principle.

Ans:

To find the derivative of a function   (  ) f(x) using the first principle (also known as the definition of the derivative), we use the following formula:

 

′ ( ) = lim ⁡ ℎ → 0 ( + ℎ ) − ( ) ℎ f  ′  (x)=lim  h→0 ​    h f(x+h)−f(x) ​

 

Let’s find the derivative of   (  ) = 2 + 2 f(x)=x  2  +2x using this principle:

 

′ ( ) = lim ⁡ ℎ → 0 ( + ℎ ) 2 + 2 ( + ℎ ) − ( 2 + 2 ) ℎ f  ′  (x)=lim  h→0 ​

 

 

3. Find 2nd order partial derivative, (∂^2 f)/(∂x^2 ), (∂^2 f)/(∂y^2 ) and (∂^2 f)/∂x∂y of f(x,y)=xe^y+ye^x.

Ans: To find the second-order partial derivatives of the function  ( , ) = + f(x,y)=xe  y  +ye  x  , we’ll start by finding the first-order partial derivatives with respect to  x and  y, and then we’ll differentiate those results again with respect to  x and  y, respectively.

 

First-order partial derivatives: ∂ ∂ ∂x ∂f ​  :

 

 

Set-IInd

 

4. Find the scalar and vector product of (A ) ⃗=(i ) ̂+(j ) ̂+3(k ) ̂ and (B ) ⃗=-2(i ) ̂+(j ) ̂+2(k ) ̂ .

Ans: To find the scalar and vector products of two vectors \( \mathbf{A} \) and \( \mathbf{B} \), we’ll first express the given vectors in terms of their components, and then apply the relevant formulas.

 

Given:

\[ \mathbf{A} = \

 

 

 

5. Apply Cramer’s rule to solve the system of equations: 3x+y+2z=3; 2x-3y-z=-3; x+2y+z=4.

Ans: To solve the given system of equations using Cramer’s rule, we’ll first express the system in matrix form and then apply the rule to find the values of \( x \), \( y \), and \( z \).

 

The system of equations can be expressed as:

 

\[ \begin{cases} 3x + y + 2z

 

 

6. Express the following complex numbers in the polar form and hence find their modulus and amplitude.

 

(i)  √3+i                      

Ans: To express a complex number in polar form, we need to represent it in terms of its modulus (magnitude) and argument (angle). Then, we can use trigonometric functions to find these values.

 

Let’s find the polar form, modulus, and argument for each complex number: